Banach’s simple Lebesgue-spectrum problem

1 · The essentials

Can one repeatedly apply the same smooth motion to a space and obtain a complete collection of independent patterns from just one measurement? The manuscript answers this question for a three-dimensional space with wraparound coordinates. Its motion preserves volume and can be run both forward and backward. Starting with one real-valued measurement whose average is zero, all its forward and backward repetitions give mutually perpendicular patterns, each of unit size. Together, these patterns can approximate every complex-valued, zero-average measurement with finite average squared size, using the corresponding notion of approximation. This is the probability-preserving form of Banach’s simple Lebesgue-spectrum problem, and the main result has a Lean formalization.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine a repeating three-dimensional room and a single brightness pattern that is carried along by a reversible motion. Every forward and backward repetition supplies another pattern, and the whole collection acts like a complete set of building blocks for all the allowed zero-average patterns. The image is imperfect because completeness and perpendicularity here concern mathematical measurements, not ordinary visual resemblance.

Why it matters

The result connects a smooth, volume-preserving motion with a precise completeness property of the measurements it generates. The completeness holds on the entire complex mean-zero space, not merely on a selected part of it. The source presents this construction as resolving the probability-preserving form of Banach’s problem within smooth dynamics.

2 · Going further

The precise setting is a $C^{\infty}$ diffeomorphism $T$ of the three-torus $T^{3}$ , preserving its standard volume. Its Koopman operator acts on observables by composition, $U f = f \circ T$ . The target is simple Lebesgue spectrum on $L_{0}^{2} (T^{3}; C)$ , the space of complex square-integrable observables with zero mean.

The repository summary makes this spectral property concrete: there is one real observable $f$ such that ${U^{n} f : n \in Z}$ is an orthonormal basis of that entire space. Orthonormality means that different iterates have zero inner product and each has norm one. Completeness means that finite complex linear combinations of these iterates approximate every observable in the space in the $L^{2}$ norm. Negative as well as positive iterates are part of the assertion.

The abstract describes the approach only as a construction of a smooth, standard-volume-preserving diffeomorphism with this property; it does not supply the construction mechanism or argument. It identifies the result as a solution of the probability-preserving form of Banach’s simple Lebesgue-spectrum problem in smooth dynamics. Only the main manuscript is supplied, so no relationship to companion manuscripts can be described from this material. The main result has a Lean formalization, but the supplied material does not specify its detailed scope.

Useful notions

Invertible smooth maps
Volume-preserving dynamics
Square-integrable functions and zero averages
Inner products and orthonormal bases
Operators acting on functions

What was known before / what is new

Before

The source identifies Banach’s simple Lebesgue-spectrum problem as the question being addressed, specifically in its probability-preserving form. It supplies no account of earlier constructions, partial results, or historical development.

New

The manuscript presents a construction of a smooth diffeomorphism of the three-torus that preserves standard volume and has simple Lebesgue spectrum on the entire mean-zero $L^{2}$ space. The repository summary specifies that the bilateral iterates of a single real observable form an orthonormal basis of the complex mean-zero space.

Limits and assumptions

The assertion concerns a constructed $C^{\infty}$ , standard-volume-preserving diffeomorphism of the three-torus, not all smooth volume-preserving systems. The spectral statement is restricted to the mean-zero space and does not include nonzero constant observables. The main result has a Lean formalization; its detailed scope is not supplied, so coverage of every formulation or claim cannot be assessed here. The abstract gives neither the construction details nor a list of questions that remain open, and no conclusion about other forms of Banach’s problem is warranted from this source.

Mini glossary

Three-torus: A three-dimensional space in which each coordinate wraps around.
Diffeomorphism: An invertible smooth map whose inverse is also smooth.
Koopman operator: The operator that transforms an observable by composing it with the underlying motion.
Mean-zero L² space: The space of functions with zero average and finite integral of their squared absolute value.
Orthonormal basis: A collection of mutually perpendicular unit vectors whose finite linear combinations approximate every vector in the space.
Simple Lebesgue spectrum: The spectral property expressed here by the existence of one observable whose forward and backward Koopman iterates form an orthonormal basis.

Keywords : Banach’s problem · Smooth dynamics · Simple Lebesgue spectrum

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Resolves the probability-preserving form of Banach's simple Lebesgue-spectrum problem within smooth dynamics. A smooth volume-preserving diffeomorphism of the standard-volume three-torus has simple Lebesgue spectrum on its entire complex mean-zero L² space: the bilateral iterates of one real observable form an orthonormal basis of that space.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/144.md

Manuscripts in this family

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A smooth three-torus diffeomorphism with simple Lebesgue spectrum

23/09/2026

To be verified

We solve the probability-preserving form of Banach's simple Lebesgue-spectrum problem in smooth dynamics. We construct a C^∞ diffeomorphism of the three-torus that preserves standard volume and whose Koopman operator has simple Lebesgue spectrum on the entire mean-zero L² space.

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